Parallel preconditioning of a sparse eigensolver

Abstract

We exploit an optimization method, called deflation-accelerated conjugate gradient (DACG), which sequentially computes the smallest eigenpairs of a symmetric, positive definite, generalized eigenproblem, by conjugate gradient (CG) minimizations of the Rayleigh quotient over deflated subspaces. We analyze the effectiveness of the AINV and FSAI approximate inverse preconditioners, to accelerate DACG for the solution of finite element and finite difference eigenproblems. Deflation is accomplished via CGS and MGS orthogonalization strategies whose accuracy and efficiency are tested. Numerical tests on a Cray T3E Supercomputer were performed, showing the high degree of parallelism attainable by the code. We found that for our DACG algorithm, AINV and FSAI are both effective preconditioners. They are more efficient than Block–Jacobi.

Introduction

An important task in many scientific applications is the computation of a small number of the leftmost eigenpairs (the smallest eigenvalues and corresponding eigenvectors) of the problem $\text{A}\text{x}\text{=λB}\text{x}$, where A and B are large, sparse, symmetric positive definite matrices. Several techniques for solving this problem have been proposed: subspace iteration [1], [15], Lanczos method [7], [11], [14], and, more recently, restarted Arnoldi–Lanczos algorithm [12], Jacobi–Davidson method [17], and optimization methods by conjugate gradient (CG) schemes [3], [9], [16].

In this paper we analyze the performance of two preconditioning techniques, when applied to an optimization method, called deflation-accelerated conjugate gradient (DACG) [8]. DACG sequentially computes a number of eigenpairs by CG minimizations of the Rayleigh quotient over subspaces of decreasing size. When effectively preconditioned, we found [4] that the efficiency of DACG well compares with that of established packages, like ARPACK [13]. In a recent work [5], the performance of DACG has also been numerically compared with that of Jacobi–Davidson method, showing that their efficiency is comparable, when a small number of eigenpairs is to be computed.

We exploit three preconditioners, Block–Jacobi, FSAI [10] and AINV [2], the latter ones falling into the class of approximate inverse preconditioners. FSAI and AINV explicitly compute an approximation, say M to A⁻¹, based on the sparse factorization of A⁻¹. Preconditioning by a product of triangular factors performs better than other techniques, mainly because the fill-in of the preconditioner is reduced. Unlike many other approximate inverse techniques, AINV and FSAI preserve the positive definiteness of the problem, which is essential in our application. The FSAI algorithm requires an a priori sparsity pattern of the approximate factor, which is not easy to provide in unstructured sparse problems. We generated the FSAI preconditioner using the same pattern as the matrix A. On the other hand, AINV is based upon a drop tolerance, ε, which in principle is more convenient for our unstructured problems. The influence of the drop tolerance has been tested. Deflation is accomplished via B-orthogonalization of the search directions. We analyzed both classical (CGS) and modified (MGS) Gram–Schmidt, and tested the accuracy and efficiency of both strategies.

We have exploited the parallel Block–Jacobi-DACG, AINV–DACG and FSAI–DACG algorithms in the solution of finite element, mixed finite element, and finite difference eigenproblems, both in two and three dimensions. A parallel implementation of the DACG algorithm has been coded¹ via a data-parallel approach, allowing preconditioning by any given approximate inverse. Ad-hoc data-distribution techniques allow for reducing the amount of communication among the processors, which could spoil the parallel performance of the ensuing code. An efficient routine for performing matrix-vector products was designed and implemented. Numerical tests on a Cray T3E Supercomputer show the high degree of parallelism attainable by the code, and its good scalability level.